An interesting and very different Tudor question!

The entire Tudor dynasty was insecure about its position from start to finish. Henry VII, of course, only came to the throne because he fought his way there (he had no real claim to it), and both he and Henry VIII had to deal with the remnants of the prior dynasty - people who likely wouldn't have ever thought of the throne before the overthrowing of Richard III, but who technically had an excellent claim on it in the absence of the original rulers. For instance, the de la Poles were the sons of Elizabeth, sixth child of Richard, Duke of York, and sister to Edward IV and Richard III; the oldest was killed on the battlefield in rebellion against Henry VII and others were executed by Henry VIII as pretenders to the throne. Edward VI had less of a problem, but Mary had to contend with the threat of Elizabeth, and Elizabeth refused to name an heir throughout her reign because she feared that she would create a rival who could unseat her. All of them had to be concerned with Plantagenet-descended subjects who appeared to be making a bid for the throne, which was treason.

Enter Henry Howard (1517-1547). The Howards were not themselves royal (unless you went several hundred years back), but Henry's great-grandfather only rose to the rank of duke because of his service to Edward IV and Richard III; more crucially, his mother was Elizabeth Stafford, the daughter of Edward Stafford, Duke of Buckingham. Stafford was the nephew of Elizabeth Woodville, wife of Edward IV (a similar not-royalty-but-related-to-royalty position as Henry VII), and was descended from the Beauforts, legitimated grandchildren of Edward III, and he himself was executed for treason in 1521 for supposedly intending to kill the king. By the 1540s, there were precious few Plantagenets left, but Henry VIII was still concerned about potential threats.

The Howards were also under suspicion for their relationship to/support of the Boleyns, as well as the tragedy of Henry's marriage to Catherine Howard, not to mention Henry's uncle becoming betrothed to Henry VIII's widowed niece, Lady Margaret Douglas, which made it look like he was angling for the throne (particularly as this was before the birth of Edward but after the illegitimizing of Mary and Elizabeth) - he was attainted and imprisoned for treason under the 1534 Succession Act, and he died in the Tower. Which leads me to another important thing to bear in mind when considering the Tudors: the law is not an unbiased institution, it's something that can be manipulated, particularly in the Tudor period. The Succession Act, which made it treason to plot to change the succession, was passed after Thomas Howard and Margaret Douglas were married. This was less important than the fact that Henry VIII felt threatened, which comes into play with our boy as well.

So Henry Howard, son of the Duke of Norfolk, was really in the middle of it. In addition to all of this familial disaster, he had been a close friend and brother-in-law to Henry FitzRoy, Henry VIII's illegitimate son with Elizabeth Blount (TOO. MANY. HENRYS), so he was well integrated into court circles - a dangerous place to be during the Reformation, when the king was assured of his power and could be cruel and fickle, and a dangerous place to be in general, as courtiers were known for backbiting and undermining each other. By 1537, after Anne's fall, Henry and his father were both the targets of accusations that they were sympathetic to the rebels of the Pilgrimage of Grace and Henry was also dealing with the anniversary of his friend FitzRoy's death; in response to an insult at this time, probably from one of the "new men" (recently ennobled men who rose due to the Reformation, whom the old families like the Howards, Nevilles, Percys, etc. resented - the Seymours were one of the "new" families), he got involved in a fight at court and was briefly exiled to Windsor Castle - a reprieve from the maiming that was mandated as punishment for such an offense. He came back to grace when the king married his cousin Catherine in 1540, but of course after her fall in 1541, he and his family were worse off than ever. In 1542, he was put in the lowly Fleet Prison due to what seems to have been another assault, this time on a man called John Leigh; he got out fairly quickly and was sent to fight against the Scots. After his return, he again got into trouble in early 1543 after going for a violent spree through the city of London with his friends, and things went even worse for him than before. The Lord Mayor instigated an investigation into his behavior, which then was picked up by a new heresy commission run by the crown. Said commission found that he had eaten meat during Lent, and then started to look into testimony about his coat of arms (which legitimately had three lions in it) looking royal and his potentially considering himself in line for the throne if something should happen to the king (it being treasonous to speculate about the king's death). In the end, he was sent back to the Fleet, but only for his antics in London - there was no apparent interest in possible royal ambitions.

After Henry got out again, he went to lead troops in war against the French and managed to stay out of trouble for a few years. Unfortunately, military losses on his part upset Henry VIII, and he was demoted and eventually recalled in 1546. That summer, his father proposed a match between his unwilling sister Mary - FitzRoy's widow - and Thomas Seymour, and Henry reportedly suggested that Mary keep protesting it until she caught the aged and unhealthy king's attention and then become his mistress so that she could rule him. He was also known to be only weakly Protestant, was in debt, and continued to alienate other courtiers with his temper and arrogance. In December of that year, he was arrested for sending a letter with vague threats toward another member of the court, and for not having attacked a castle on the continent as a commander when he could have. His father was also arrested, though on no specific charge. An investigation ensued, with the duke's mistress and Mary being interrogated, and Mary confessed that Henry had been messing around with his armorial bearings, adding the Stafford crest to it (which was not allowed, since Edward Stafford had been attainted and therefore his titles and such had been forfeit to the crown) as well as a heraldic cap of maintenance, something reserved for royalty in England; others gave testimony of other, lesser failings, and popular opinion at court turned very much against him, with rumors of conspiracies and assassinations flying.

Finally, Henry was charged under that same Succession Act that saw his uncle die in the Tower, as wanting to tamper with the succession, i.e. by adding himself to it. Investigations had found that while he was experimenting with new arms, he had indeed added those of Edward the Confessor into a quarter of his own, which he had the right to do via an ancestor that had been given the privilege by Richard II. However, his deposition was shown to the king with an error in it that made him appear to be lying, and alongside the royal lineage that he had bragged about, it was possible to justify him as attempting to become royal. In January 1547, his father signed a confession of his son's guilt, and Henry was tried for treason. The charge itself asserted that anyone using these arms was guilty and the king directed the jury not to listen to arguments justifying their use, so it was a foregone conclusion. The one mercy he was granted was to be beheaded rather than suffering the fate of being hanged, drawn, and quartered as was normally the punishment for traitors.

If you want to read more about Henry Howard, I would suggest Henry Howard: Henry VIII's Last Victim, by Jessie Childs.